Smooth entropy in axiomatic thermodynamics Mirjam Weilenmann1, Lea Kr¨amerGabriel2, Philippe Faist3, and Renato Renner2 1Department of Mathematics, University of York, Heslington, York, YO10 5DD, UK 2Institute for Theoretical Physics, ETH Zurich, 8093 Switzerland 3Institute for Quantum Information and Matter, California Institute of Technology, Pasadena 91125, USA July 23, 2018 Abstract Thermodynamics can be formulated in either of two approaches, the phenomenological approach, which refers to the macroscopic properties of systems, and the statistical approach, which describes systems in terms of their microscopic constituents. We establish a connection between these two approaches by means of a new axiomatic framework that can take errors and imprecisions into account. This link extends to systems of arbitrary sizes including microscopic systems, for which the treatment of imprecisions is pertinent to any realistic situation. Based on this, we identify the quantities that characterise whether certain thermodynamic processes are possible with entropy measures from information theory. In the error-tolerant case, these entropies are so-called smooth min and max entropies. Our considerations further show that in an appropriate macroscopic limit there is a single entropy measure that characterises which state transformations are possible. In the case of many independent copies of a system (the so-called i.i.d. regime), the relevant quantity is the von Neumann entropy. 1 Introduction The thermodynamic behaviour of macroscopic systems is traditionally described according to either of two theories: we can take a phenomenological approach that refers to macroscopic quantities such as the volume and the pressure of a system in which the possible thermodynamic processes are constrained via the traditional laws of thermodynamics, or we can take a statistical approach that begins with a microscopic description of the system, for instance by describing the motion of individual particles, and then infers the system's corresponding macroscopic properties via statistical and typicality considerations. The two theories are fundamentally different, each referring to quantities that are not defined within the other, but at the same time they are known to lead to consistent descriptions of the behaviour of thermodynamic systems in equilibrium. The two approaches are usually related by studying specific examples and connecting the corresponding physical quantities, such as identifying the Boltzmann entropy with the thermodynamic entropy in the case of an isolated ideal gas or a spin chain. In standard textbooks, such identifications are largely carried out based on the properties these quantities display: For instance, the Boltzmann entropy and the thermodynamic entropy are both extensive, and their derivatives with respect to the total energy are both equal to the inverse temperature. While the correspondence between these approaches clearly holds at a very universal level, a precise and general connection is hindered by the very different underlying physical frameworks assumed in either arXiv:1807.07583v1 [quant-ph] 19 Jul 2018 approach. The aim of this chapter is to describe the phenomenological and the statical approach on the same footing, i.e., using the same framework. Rather than focusing on derived quantities such as the entropy, we connect the two approaches on the level of their basic underlying structure. This structure is rooted in identifying the possible processes that can occur spontaneously. While it is rather naturally suited to phenomenological thermodynamics, and can be interpreted as an abstract version thereof, it also applies to the microscopic realm pertinent to the statistical approach. In this abstract structure, whether or not a process may occur is characterized by a specific quantity|which is called, by extension, an entropy function. In the phenomenological approach, it coincides with the thermodynamic entropy and naturally characterises the possible adiabatic processes. It turns out that, in order to relate the phenomenological to the statistical approach, it is critical to take small perturbations into account. One must allow a degree of imprecision in processes in order to obtain physical results that do not depend on unobservable features of the quantum state, such as distribution tails (for instance, when 1 Smooth entropy in axiomatic thermodynamics July 23, 2018 isothermally compressing a gas, one typically ignores the overwhelmingly unlikely event in which all particles con- spire to hit against the piston, which would require more invested work). We show that the abstract structure of possible processes extends naturally to the case where imprecisions are to be taken into account. In an appropriate macroscopic limit, such as by considering a large system composed of many independent particles, we recover usual phenomenological thermodynamics. That is, we identify a class of macroscopic states whose structure of possible processes coincides with that of the usual phenomenological approach. Hence, beyond the fact that both the phe- nomenological approach and the statistical approach can be phrased within the same type of framework, the former can be obtained from the latter on an abstract level, thus drawing a robust and general connection between them that transcends individual examples. To establish our results, we rely on a resource theoretic formulation of thermodynamics. There are several approaches to this, depending on which operations are considered to be the free operations of the resource theory. The most widespread approach relies on thermal operations [1{3], which are reviewed in [4]. Instead, we consider adiabatic processes here, which are central to Lieb and Yngvason's axiomatic approach to phenomenological thermodynam- ics [5{10]. Their approach follows a long tradition of formulating thermodynamics axiomatically [5{20], which has also been continued by recent work [21{24]. In an adiabatic process, a system interacts with a weight that can per- form or extract work from the system without changing the environment. Contrary to the thermal operations, these processes do not impose any constraint regarding the conservation of energy. Instead, they forbid the equilibration with a reservoir, which would result in a change of the state of the system's environment. We introduce Lieb and Yngvason's axiomatic framework for thermodynamics in Section 2 and relate the notion of an adiabatic process to the statistical picture in Section 3. Based on this, we present an axiomatic relation between the entropy measures relevant for thermodynamics in the phenomenological and in the statistical approach. This is essentially a summary of results derived in Ref. [25]. Furthermore, we introduce a novel, error-tolerant axiomatic framework that allows us to realistically describe sys- tems at any scale, including microscopic and macroscopic systems. (Previous approaches were typically concerned solely with macroscopic systems and did not take errors into account.) Our framework contributes to the current development of pushing resource theories towards more realistic regimes, which has been initiated through work on probabilistic transformations [26] and finite size effects [27; 28] and which has recently also led to the study of im- precisions in specific resource theories [29{33]. Similar to our consideration of Lieb and Yngvason's work, we present our new framework first phenomenologically in Section 4 and then from the statistical viewpoint in Section 5. This allows us to relate the quantities that characterise whether there exists an error-tolerant adiabatic process between certain states to smooth min and max entropies, which are known from single-shot information theory [34{36]. In the limit of large systems, our error-tolerant framework furthermore recovers the structure of a resource theory for macroscopic thermodynamics. Specifically, there is a single entropy function that specifies whether a transformation between different equilibrium states of a macroscopic system is possible. In the spirit of the chapter, we present these results first according to the phenomenological approach in Section 6, followed by the statistical perspective in Section 7. The latter viewpoint allows us to recover the von Neumann entropy as the quantity that characterises the behaviour of so-called i.i.d. states under error-tolerant adiabatic processes. Through this consideration of macroscopic systems in equilibrium we relate our error-tolerant framework to the framework for thermodynamics introduced by Lieb and Yngvason. Our elaborations in Sections 4 to 7 regarding the error-tolerant axiomatic framework and its macroscopic limits are based on [33]. 2 Axiomatic framework for phenomenological thermodynamics In phenomenological thermodynamics, the state of a system in equilibrium is usually described in terms of a few real and positive parameters. For instance, the state of a gas in a box is often given in terms of its internal energy, U, and its volume, V . Such an equilibrium state X = (U; V ) lives in the space of all equilibrium states of that system, denoted ΓEQ. The description of the state of a system out of equilibrium is more involved; such states live in a larger state space Γ ⊇ ΓEQ. Thermodynamics is a resource theory, a perspective that is implicit in the axiomatic framework proposed by Lieb and Yngvason to derive the second law [5{9]. At the core of this framework lie the adiabatic processes. They are defined in [5{9] as those